In Unit 5, before you explore the unit you must have previous knowledge of what a Cartesian Plane is. Also, what the word variable means. Mainly because there will be many variables in every problem of both solving linear equations and graphing linear equations. By the end of this unit you will know what y=mx+b stands for and how to use it. Secondly, you will be able to place to points on a line through a chart then find it’s full equation, use slope formula/find the slope of a line and rate of change(including parallel and perpendicular lines) to find the equation of a line. With each of the previous skills you will have at your side by the end of the unit you will also have many key pieces of new information. Some key pieces that are presented in this unit are undefined lines and find the equation of a line by just viewing 2 points. While you are developing each skill there are misconceptions you can come across such as thinking that every solution is a function, which can be very untrue. This is an example created word problem that exists in this unit.
A chart is designed for a engineer to build a robot in 3 days. He has all of this x values filled in on his chart in chronological order of: -1,0,1,2,3. He doesn't know his calculations yet. But he also doesn't know his y values. He wrote down this equation: y=5x^2+4. Fill in the rest of his uncompleted work.
In this word problem it is asking for you to design a chart and make the calculations for it. First place the values of y under the y column. It can become confusing where there is a 5 next to the x and it is square. This means that whatever the value of x is, it is squared. Then you have to continue the steps for solving for y. The chart should look like fig. 1 when you are finished
fig.1. y=mx+b; m= slope (Rise over Run) b= y-intercept (This is where your first point will be placed along the y-axis)
X
Calc
Y
-1
y=5(-1)^2+4
9
0
y=5(0)^2+4
4
1
y=5(1)^2+4
9
2
y=5(2)^2+4
24
3
y=5(3)^2+4
49
Each x value also has a Y value. Through the calculation box we distributed the variables given for x in the x place of the equation and then found out what y is equal to. We can also check this by plugging in the y variables in the y place. With the x variables also in it’s place. Some exact examples that can appear throughout Mathematics are below. (By Alicia Jones)
In a Geo./Alg. 2:
5 x - 6 = 3 x - 8
In this problem you have to make sure to isolate the variable for the unknown variable.
Steps to solve this problem: First: In this first step of this equation you have to subtract 3 from it’s like term. A like term is a number that has the same variables, coefficients or power(exponential power). In this case you would subtract 3x from 5x. Your 3x and -3x should cancel out. A quick thing to remember is that you cannot subtract a term with a variable attached from a normal number. They AREN’T like terms. When you have solved for a new difference on the left side of the equation you should have.
1.) 5x-6=3x-8 -3x -3x (subtract 3x on both sides) 2x-6=-8 Secondly: You now have to get only one number on both side of the equal sign. One side can have a variable plus coefficient and the other side can have a normal number. Now to achieve this goal you have to get rid of one of the two numbers which are on the left side of the equal sign. This will fall under the category of isolation. Begin by adding 6 on both side of the equation then divide to find out the value of x.
1.) 2x-6=-8 +6 +6 2x = -2 2 [ x=-1 ] Lastly: You can check your answer by plugging in the value for x back into the original equation.
Check: 5 x - 6 = 3 x - 8 5(-1)-6=3(-1)-8 -11 = -11 > It checks! In conclusion, the answer checks because on both sides of the equal sign it is equivalent to the other.
An exact example from the SAT’s :
2x+4y=50 3x+5y=66 In this problem there are two equations which is something we haven’t reviewed over very well yet. So this problem will be a bit tedious and challenging.
Steps to solve this problem: First: In the first equation we can solve for x because each value can has a GCF of 2. Not 2x because 4y nor 50 has a variable of x behind it. This will lead us to rearranging the equation from the original state. You will subtract 4y on the other side of the equal sign so you can later divide easier and also to get 2x by itself. Remember that you cannot subtract 4y from 50 because they are not like terms. So you will simply just be removing 4y from one side of the equation to the other. Then divide by 2x to simplify.
1.) 2x+4y=50 -4y-4y 2x=50-4y
2.) 2x=50-4y 2
X value: [ x=25-2y ]
Secondly: After you have a value for x you will distribute this value for ex back into either or of two given equations to now solve for y. This is quite challenging because we have to estimate for a number that is close to 50 but not exactly at 50. So when we decide to fill in the x variable we can still multiply it by the coefficient, which is 2 and have a sum of 50. The number that can take the place of y is 9. Next you have to subtract to get only one number on the other side of the equal sign. This other number cannot have any variables next to it. Therefore, we have to subtract 36 from 50. Your difference will be 14. Then divide by 2 on both sides of the equal sign and find out what x equals. 1.) 2x+4y=50 2x+4(9)=50 2x+(36)=50 [ x=7]
2.)
2x+36=50 -36 -36 2x=14
[ y=9 ]
To check this, all you have to do is plug each variable back into the equation. However, you shouldn’t use the same equation use used to find out what both variables were. Instead use the second given equation in the problem. Checking: 3x+5y=66 3(7)+5(9)=66 > It checks! In conclusion both of your variables for the previous examples is: (7,9)
In a Real-Life situation: A cab company charges a $4 boarding rate in addition to its meter which is $ ¾ for every mile. What is the equation of the line that represents this cab company’s rate?
Answer: y=3/4x+4
It would be “y=¾x+4” because the cab company charges the up front fee, just to board, $4.00. This becomes you Y-intercept, in other words this is where your linear equation would begin on a graph. Then you would graph up 3 and over 4. This represents the ¾ for every mile on the meter.
Steps to solve this problem: First: Now to graph this you would simply start the first step by finding the y-intercept in your equation. Then placing that first point on the graph. y-intercept= (0,4) Secondly: After you completed the first step you now have to look for the slope or in other words the m in y=mx+b of the equation that you setup. From your y-intercept point you should graph up and over. in this case you should graph like the following. Then find the line of the equation. Slope= ¾ - Up 3 ; Over 4 Secondly: You would look at your m variable place and then find the slope given. Position the slope in the appropriate place and then continue to graph until your line is formed.
Create Your own / Created example problem:
1.) The rental car company charges a $15.00 car fee for any model car being rented for a prom. In addition to the car fee there is a $2.00 cost per every 4 miles driven. Mary’s mom want’s to rent a car for Mary’s prom. Set up the correct linear equation that Mary’s mom will have to write out. Then graph.
Answer: y=2/4x+15 This is the correct linear equation because if the rental car company is charging a $15.00 upfront fee for any car, this is where your base charges begin. In other words, your y-intercept is formed. Secondly, your variable of m should be 2/4 because every 4 miles there is a $2.00 charge.
Steps to solve this problem: First: You have to find the y-intercept or where the base charge is located at in the word problem. When you find that then you have to designate it on the graph as the following coordinate point: y-intercept: (0,15) Secondly: Next, there is the slope that you must graph. Start at your y-intercept and graph up and over. So from the coordinate (0,15) you have to graph up Then find the line of your equation. Slope=2/4; Up 2 and Over 4.
In Unit 5, before you explore the unit you must have previous knowledge of what a Cartesian Plane is. Also, what the word variable means. Mainly because there will be many variables in every problem of both solving linear equations and graphing linear equations. By the end of this unit you will know what y=mx+b stands for and how to use it. Secondly, you will be able to place to points on a line through a chart then find it’s full equation, use slope formula/find the slope of a line and rate of change(including parallel and perpendicular lines) to find the equation of a line. With each of the previous skills you will have at your side by the end of the unit you will also have many key pieces of new information. Some key pieces that are presented in this unit are undefined lines and find the equation of a line by just viewing 2 points. While you are developing each skill there are misconceptions you can come across such as thinking that every solution is a function, which can be very untrue. This is an example created word problem that exists in this unit.
A chart is designed for a engineer to build a robot in 3 days. He has all of this x values filled in on his chart in chronological order of: -1,0,1,2,3. He doesn't know his calculations yet. But he also doesn't know his y values. He wrote down this equation: y=5x^2+4. Fill in the rest of his uncompleted work.
In this word problem it is asking for you to design a chart and make the calculations for it. First place the values of y under the y column. It can become confusing where there is a 5 next to the x and it is square. This means that whatever the value of x is, it is squared. Then you have to continue the steps for solving for y. The chart should look like fig. 1 when you are finished
fig.1.
y=mx+b;
m= slope (Rise over Run)
b= y-intercept (This is where your first point will be placed along the y-axis)
Each x value also has a Y value. Through the calculation box we distributed the variables given for x in the x place of the equation and then found out what y is equal to. We can also check this by plugging in the y variables in the y place. With the x variables also in it’s place.
Some exact examples that can appear throughout Mathematics are below. (By Alicia Jones)
In a Geo./Alg. 2:
5 x - 6 = 3 x - 8
In this problem you have to make sure to isolate the variable for the unknown
variable.
Steps to solve this problem:
First: In this first step of this equation you have to subtract 3 from it’s like term. A like term is a number that has the same variables, coefficients or power(exponential power). In this case you would subtract 3x from 5x. Your 3x and -3x should cancel out. A quick thing to remember is that you cannot subtract a term with a variable attached from a normal number. They AREN’T like terms. When you have solved for a new difference on the left side of the equation you should have.
1.)
5x-6=3x-8
-3x -3x (subtract 3x on both sides)
2x-6=-8
Secondly: You now have to get only one number on both side of the equal sign. One side can have a variable plus coefficient and the other side can have a normal number. Now to achieve this goal you have to get rid of one of the two numbers which are on the left side of the equal sign. This will fall under the category of isolation. Begin by adding 6 on both side of the equation then divide to find out the value of x.
1.)
2x-6=-8
+6 +6
2x = -2
2 [ x=-1 ]
Lastly: You can check your answer by plugging in the value for x back into the original equation.
Check: 5 x - 6 = 3 x - 8
5(-1)-6=3(-1)-8
-11 = -11 > It checks!
In conclusion, the answer checks because on both sides of the equal sign it is equivalent to the other.
An exact example from the SAT’s :
2x+4y=50
3x+5y=66
In this problem there are two equations which is something we haven’t reviewed over very well yet. So this problem will be a bit tedious and challenging.
Steps to solve this problem:
First: In the first equation we can solve for x because each value can has a GCF of 2. Not 2x because 4y nor 50 has a variable of x behind it. This will lead us to rearranging the equation from the original state. You will subtract 4y on the other side of the equal sign so you can later divide easier and also to get 2x by itself. Remember that you cannot subtract 4y from 50 because they are not like terms. So you will simply just be removing 4y from one side of the equation to the other. Then divide by 2x to simplify.
1.)
2x+4y=50
-4y-4y
2x=50-4y
2.)
2x=50-4y
2
X value: [ x=25-2y ]
Secondly: After you have a value for x you will distribute this value for ex back into either or of two given equations to now solve for y. This is quite challenging because we have to estimate for a number that is close to 50 but not exactly at 50. So when we decide to fill in the x variable we can still multiply it by the coefficient, which is 2 and have a sum of 50.
The number that can take the place of y is 9. Next you have to subtract to get only one number on the other side of the equal sign. This other number cannot have any variables next to it. Therefore, we have to subtract 36 from 50. Your difference will be 14. Then divide by 2 on both sides of the equal sign and find out what x equals.
1.)
2x+4y=50
2x+4(9)=50
2x+(36)=50
[ x=7]
2.)
2x+36=50
-36 -36
2x=14
[ y=9 ]
To check this, all you have to do is plug each variable back into the equation. However, you shouldn’t use the same equation use used to find out what both variables were. Instead use the second given equation in the problem.
Checking:
3x+5y=66
3(7)+5(9)=66 > It checks!
In conclusion both of your variables for the previous examples is: (7,9)
In a Real-Life situation:
A cab company charges a $4 boarding rate in addition to its meter which is $ ¾ for every mile. What is the equation of the line that represents this cab company’s rate?
Answer: y=3/4x+4
Steps to solve this problem:
First: Now to graph this you would simply start the first step by finding the y-intercept in your equation. Then placing that first point on the graph.
y-intercept= (0,4)
Secondly: After you completed the first step you now have to look for the slope or in other words the m in y=mx+b of the equation that you setup. From your y-intercept point you should graph up and over. in this case you should graph like the following. Then find the line of the equation.
Slope= ¾ - Up 3 ; Over 4
Secondly: You would look at your m variable place and then find the slope given. Position the slope in the appropriate place and then continue to graph until your line is formed.
Create Your own / Created example problem:
1.) The rental car company charges a $15.00 car fee for any model car being rented for a prom. In addition to the car fee there is a $2.00 cost per every 4 miles driven. Mary’s mom want’s to rent a car for Mary’s prom. Set up the correct linear equation that Mary’s mom will have to write out. Then graph.
Answer: y=2/4x+15
This is the correct linear equation because if the rental car company is charging a $15.00 upfront fee for any car, this is where your base charges begin. In other words, your y-intercept is formed. Secondly, your variable of m should be 2/4 because every 4 miles there is a $2.00 charge.
Steps to solve this problem:
First: You have to find the y-intercept or where the base charge is located at in the word problem. When you find that then you have to designate it on the graph as the following coordinate point:
y-intercept: (0,15)
Secondly: Next, there is the slope that you must graph. Start at your y-intercept and graph up and over. So from the coordinate (0,15) you have to graph up Then find the line of your equation.
Slope=2/4; Up 2 and Over 4.